Source: http://www.abarim-publications.com/Meaning/sheminith.html

🔼The word sheminith in the Bible

The word sheminith is a musical term which occurs three times in the Bible. When king David transported the Ark of the Covenant from the house of Obed-edom to the tabernacle at Jerusalem, he had organized for several groups of Levite musicians to come along and make merriment. These musicians either sang, played harp, lyre or cymbals or simply cheered. The men with the harps had their instruments על־עלמות ('al-alamoth; 1 Chronicles 15:20), while the lyre players had their instruments על־השמינית ('al-hasheminith; 1 Chronicles 15:21).

The word על ('al) means on or towards or in this context probably "set to" and the words alamoth and sheminith are musical terms. It's not wholly clear what kind of musical term these are. They may describe the type of instrument, but they may also have to do with the tuning of the instrument. Because the harpists and lyrists appear to have played simultaneously during the procession, these tunings or types of instrument would have had to be harmonious.

The phrase על־השמינית ('al-hasheminith) occurs twice more, namely in the headings of Psalm 6:1 and 12:1. Psalm 12 mentions no instruments but in Psalm 6 our phrase is associated with the verb נגן (nagan), meaning to play a stringed instrument. This seems to suggest that both harps and lyres could exist as sheminith.

🔼Etymology of the word sheminith

The word sheminith appears to derive from the root שמן II (shmn II), which has to do with the number eight:

🔼Sheminith meaning

It's ultimately unclear what the word sheminith means. It may mean Oily, after שמן I (shmn I), and perhaps be somehow associated with royalty (David played his signature harp, after all). But most, if not all scholars derive it from the adjective שמני (shemini) or שמנית (sheminit), meaning Eighth.

Some scholars insist that the sheminith was an eight-stringed harp or lyre, whereas others believe that any instrument tuned to a sheminith was an instrument that covered a whole octave of notes. Very old Sumerian tablets indicate that the Semites possessed thorough knowledge of musical theory.

But on the other hand, the word sheminith may very well be the title (first word) of some famous song, whose catchy tune carried several subsequent compositions. We simply don't know.